Staff: Mentor

Question 2
In the above text from Anderson and Feil we read the following:
" ... ... Because a field satisfies multiplicative cancellation, no two of these elements are the same .. ... "
Can someone please demonstrate exactly why it follows that no two of the elements of ##S## are the same .. ... "

It's also obvious. If we assumed ##[x\cdot n]=[x\cdot m]## then what would this mean? You can use all field operations and again the lack of zero divisors. In general, the question to be answered is: why do fields allow multiplicative cancellations? The reason doesn't require this special field, so you may as well prove: ##a\cdot m = b \cdot m \Longrightarrow a = b## for ##a,b \neq 0## of course.